Problem: Let $f(x)=\cos(x)+2x$. Where does $f$ have critical points? Choose all answers that apply: Choose all answers that apply: (Choice A) A $x=0$ (Choice B) B $x=1$ (Choice C) C $x=\pi$ (Choice D) D $f$ has no critical points.
Answer: A critical point of $f$ is a point in the domain of $f$ where the derivative is either equal to zero or undefined. So in order to find the critical points of $f$, let's find its derivative. $\begin{aligned} f'(x)&=\dfrac{d}{dx}\left[ \cos(x)+2x \right] \\\\ &=-\sin(x)+2 \end{aligned}$ Now let's look for $x$ -values where $f'$ is zero or undefined. $-\sin(x)+2=0$ has no solution, so $f'$ never equals $0$. $-\sin(x)+2$ is never undefined, so $f'$ is never undefined. In conclusion, $f$ has no critical points.